Question 30
Given $0\le x_0\lt1,$ let $$
x_n= \begin{cases} 2x_{n-1},&\text{if }2x_{n-1}\lt1,\\ 2x_{n-1}-1,&\text{if }2x_{n-1}\ge1 \end{cases}
$$ for all integers $n\gt0.$ For how many $x_0$ is it true that $x_0=x_5?$
x_n= \begin{cases} 2x_{n-1},&\text{if }2x_{n-1}\lt1,\\ 2x_{n-1}-1,&\text{if }2x_{n-1}\ge1 \end{cases}
$$ for all integers $n\gt0.$ For how many $x_0$ is it true that $x_0=x_5?$